where the parameters $\lambda_0$ and $\lambda_1$ define the market price of risk in a so called essentially affine model as proposed by Duffie and Kan, such that the market price of risk $\Lambda(t)=\lambda_0 + \lambda_1 \cdot r(t)$. For now, I considered only the case $\lambda_1 = 0$.

The state price deflator $\Pi(t)$ will then be the solution to the following SDE:

Now, having implemented a simulation loop for these processes, I would like to verify that the theoretical affine model zero coupon bond price

$$P(t,T) = e^{A(T)-B(T)r(t)}$$

coincides with my simulations, eg. I would like to verify that

$$P(0, 10)=E^Q\left[e^{-\int_0^{10}r(t)\mathrm{d}t}\right]=E^P\left[\frac{\Pi(10)}{\Pi(0)}\cdot 1\right]=E^P\left[\Pi(10)\right].$$
I do this by simulating $r$ under the risk neutgral $\mathbb{Q}$-measure for the exponential integral, and this coincides with the theoretical affine model zero coupon bond price $P(0,10)=e^{A(10)-B(10)r(0)}$.

Then, as to my understanding of state price deflators, I would need to use the $r$-process simulated under the real world $\mathbb{P}$-measure in order to calculate the zero coupon bond price using SPD, ie.